addition to finding lines (axes) of symmetry, you can also look for points
point of symmetry is a point that represents a "center"
of sorts for the figure. For any line that you draw through the
point of symmetry, if this line crosses the figure on one side
of the point, the line will also cross the figure on the other
side of the point, and at exactly the same distance from the point
instance, a figure-eight has a point of symmetry in the middle,
where the lines cross (shown by the blue dot):
an hyperbola, the center is the point of symmetry:
you can see from the hyperbola, a point of symmetry doesn't have
to be a point on the figure; it can lie outside the figure or
graph; in this case, the point of symmetry happens to be the origin.
can also view points of symmetry as being points about which you can rotate
the shape 180°, as shown below.
and points of symmetry can be anywhere on the plane. Points of symmetry
do not have to be the origin; lines of symmetry do not have to correspond
to either axis.
of symmetry: not at the origin
of symmetry: not along either axis
The points and lines of symmetry
do not even have to touch the figure, and lines of symmetry do not
have to be vertical:
of symmetry: not on the ellipse
line of symmetry: not touching the hyperbola
It should be noted, however,
that the lines of symmetry for the ellipse and the vertical
line of symmetry for the hyperbola will touch their respective figures.
Whether or not particular lines and points of symmetry will touch their
figures will vary from figure to figure.